Anticoncentration and state design of random tensor networks

Abstract

We investigate quantum random tensor network states where the bond dimensions scale polynomially with the system size, $N$. Specifically, we examine the delocalization properties of random Matrix Product States (RMPS) in the computational basis by deriving an exact analytical expression for the Inverse Participation Ratio (IPR) of any degree, applicable to both open and closed boundary conditions. For bond dimensions $\chi \sim \gamma N$, we determine the leading order of the associated overlaps probability distribution and demonstrate its convergence to the Porter-Thomas distribution, characteristic of Haar-random states, as $\gamma$ increases. Additionally, we provide numerical evidence for the frame potential, measuring the $2$-distance from the Haar ensemble, which confirms the convergence of random MPS to Haar-like behavior for $\chi \gg \sqrt{N}$. We extend this analysis to two-dimensional systems using random Projected Entangled Pair States (PEPS), where we similarly observe the convergence of IPRs to their Haar values for $\chi \gg \sqrt{N}$. These findings demonstrate that random tensor networks with bond dimensions scaling polynomially in the system size are fully Haar-anticoncentrated and approximate unitary designs, regardless of the spatial dimension.

References in corpus (8)

• Supplementary information for "Quantum supremacy using a programmable superconducting processor"
• The density-matrix renormalization group in the age of matrix product states
• Classical simulation of infinite-size quantum lattice systems in two spatial dimensions
• Evenly distributed unitaries: on the structure of unitary designs
• Minimally Entangled Typical Thermal State Algorithms
• Scrambling in Random Unitary Circuits: Exact Results
• Efficient Basis Formulation for (1+1)-Dimensional SU(2) Lattice Gauge Theory: Spectral calculations with matrix product states
• Participation spectroscopy and entanglement Hamiltonian of quantum spin models

Cited by in corpus (6)

• Fermionic Magic Resources of Quantum Many-Body Systems
• Anticoncentration and State Design of Doped Real Clifford Circuits and Tensor Networks
• Scalable projected entangled-pair state representation of random quantum circuit states
• Anticoncentration in Clifford Circuits and Beyond: From Random Tensor Networks to Pseudo-Magic States
• Pseudoentanglement from tensor networks
• Anti-concentration is (almost) all you need